911 
eats sin at, 1 -1 Sin at, (1-15) 
@ = ° rm = 
‘a cos at, = 1’ ‘n ry cos at, = 1 
Perform the following triangular substitution: 
h = V2(1 = cos wt 
( 0) sin ot, = sin at, 
sin ot, h 
atm (I-16) 
cos at, ~ 1 cos ot, = baie “2 lt 
Combining Eqae (I-14) and (I-16): 
5 = [eos ®t, (cos wt, = 1) + sin® at, = cos wt, + 1] 
In = ae V1 = cos at, (I-17) 
Since the response to a step function is 2F/k (Eqe I-11), the relative 
response 
Y= 00707 V1 = cos wt, (I-18) 
Consider Eqe (I-15): 
sin at 
tan ot, = ———_°— (I-15) 
cos at, - 1 
x sin x 
s + = 
ince co a Leese 
ot a ot 
tan wt, = - cot —— = tan| — + —2 
eats Fy «([5 a 
Therefore 
ot, = 2+ Sto (I-19) 
F) 2 
Figure I-1 consists ox generalized plots giving ¥ and wt, /n as 
functions of at./ne 
=%6= 
